Re: Polylog equations
- To: mathgroup at smc.vnet.net
- Subject: [mg85012] Re: Polylog equations
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Thu, 24 Jan 2008 04:38:58 -0500 (EST)
- References: <fn21oa$in5$1@smc.vnet.net>
Valeri Astanoff <astanoff at gmail.com> wrote:
> Good day,
>
> Given these polylog equations:
>
> In[1]:= Assuming[0 < x < 1,
> Solve[PolyLog[3/2,x]==y && PolyLog[5/2,x]==z,z,x]]
>
> Out[1]= {{}}
>
> what is the best way to get z(y)?
I haven't seen my previous reply or any replies from others appear in the
newsgroup yet. But Valeri and I have had some correspondence by private
email. From that, here's some information that may interest others.
According to Valeri, Albert Einstein gave an approximation for z(y) in
writing about quantum theory of ideal gases: z(y) is approximately
y - 0.1768 y^2 - 0.0034 y^3 - 0.0005 y^4
But those appear to be merely the first four terms of a Maclaurin series,
which I found easily with Mathematica:
In[7]:= Normal[
PolyLog[5/2, InverseSeries[Series[PolyLog[3/2, x], {x, 0, 4}], y]]]
Out[7]= y - y^2/(4*Sqrt[2]) + (1/8 - 2/(9*Sqrt[3]))*y^3 +
(-(3/32) - 5/(32*Sqrt[2]) + 1/(2*Sqrt[6]))*y^4
In[8]:= N[%]
Out[8]= y - 0.176777 y^2 - 0.00330006 y^3 - 0.000111289 y^4
and so it would seem that, if Einstein had intended to give the first four
terms of that Maclaurin series, either his numerical work was not quite
right or I (or Mathematica) made a mistake.
Valeri also noted that z(Zeta[3/2]) is Zeta[5/2], which is about 1.34148...
For comparison, using Einstein's approximation, one gets 1.3219, while
using four terms of the actual Maclaurin series, one gets 1.34194...
David W. Cantrell